On the other hand, in the chapter 5.6 you state that the movement is described by the geometric Brownian motiondS_t = mu*S_t*dt + sigma*S_t*dWt.

How to reconcile the two? Do I understand correctly, that in the meantime you silently changed the definition of mu (which in 3.7 does not stand for the actual growth as there is also the "jagging condribition" 1/2 sigma^2).